Using Theorem 1 (specifying x0=1) , a binary function ^ on N can be constructed such that:

1. x^0=12. x^(y+1) = x^y * x

Using this a definition, the usual Laws of Exponents can be derived:

1. The Product of Powers Rule: x^y * x^z = x^(y+z)2. The Power of a Power Rule: (x^y)^z = x^(y*z)3. The Power of a Product Rule: (x*y)^z = x^z * y^z

However, using Theorem 1 (specifying x0=0 instead) , another binary function ^ on N can also be constructed such that:

1. 0^0=02. x^0=1 for x=/=03. x^(y+1) = x^y * x

The two functions are almost identical, differing only on the value of 0^0 itself (see Theorem 2).

How to choose? For the obvious reasons, 0^0=1 is much more convenient. But the fact is, there does not seem to be any logically compelling reason to choose either alternative. The only real alternative seems to be to leave 0^0 undefined.